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1. Write down a unit vector in XY-plane, making an angle of \(30^°\) with the positive direction of x-axis.

2. Find the scalar components and magnitude of the vector joining the points \(P(x_1,y_1,z_1)\) and \(Q(x_2,y_2,z_2)\).

3. A girl walks 4 km towards west, then she walks 3 km in a direction \(30^°\) east of north and stops. Determine the girl’s displacement from her initial point of departure.

4. If \(\vec{a}=\vec{b}+\vec{c}\), then is it true that \(\left|\vec{a}\right|=\left|\vec{b}\right|+\left|\vec{c}\right|\)? Justify your answer.

5. Find the value of x for which \(x\left(\hat{i}+\hat{j}+\hat{k}\right)\) is a unit vector.

6. Find a vector of magnitude 5 units, and parallel to the resultant of the vectors

\(\vec{a}=2\hat{i}+3\hat{j}-\hat{k}\) and \(\vec{b}=\hat{i}-2\hat{j}+\hat{k}\).

7. If \(\vec{a}=\hat{i}+\hat{j}+\hat{k}\), \(\vec{b}=2\hat{i}-\hat{j}+3\hat{k}\) and \(\vec{c}=\hat{i}-2\hat{j}+\hat{k}\), find a unit vector parallel to the vector \(2\vec{a}-\vec{b}+3\vec{c}\).

8. Show that the points A(1,–2,–8), B(5,0,–2) and C(11,3,7) are collinear and find the ratio in which B divides AC.

9. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are \(\left(2\vec{a}+\vec{b}\right)\) and \(\left(\vec{a}-3\vec{b}\right)\) externally in the ratio 1:2. Also, show that P is the mid point of the line segment RQ.

10. The two adjacent sides of a parallelogram are \(2\hat{i}-4\hat{j}+5\hat{k}\) and \(\hat{i}-2\hat{j}-3\hat{k}\). Find the unit vector parallel to its diagonal. Also, find its area.

11. Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are \(\frac{1}{\sqrt3},\frac{1}{\sqrt3},\frac{1}{\sqrt3}\).

12. Let \(\vec{a}=\hat{i}+4\hat{j}+2\hat{k}\), \(\vec{b}=3\hat{i}-2\hat{j}+7\hat{k}\) and \(\vec{c}=2\hat{i}-\hat{j}+4\hat{k}\). Find a vector \(\vec{d}\) which is perpendicular to both \(\vec{a}\) and \(\vec{b}\), and \(\vec{c}.\vec{d}=15\).

13. The scalar product of the vector \(\hat{i}+\hat{j}+\hat{k}\) with a unit vector along the sum of vectors \(2\hat{i}+4\hat{j}-5\hat{k}\) and \(\lambda\hat{i}+2\hat{j}+3\hat{k}\) is equal to one. Find the value of \(\lambda\).

14. If \(\vec{a},\vec{b},\vec{c}\) are mutually perpendicular vectors of equal magnitudes, show that the vector \(\vec{a}+\vec{b}+\vec{c}\) is equally inclined to \(\vec{a},\vec{b}\) and \(\vec{c}\).

15. Prove that \(\left(\vec{a}+\vec{b}\right).\left(\vec{a}+\vec{b}\right)=\left|\vec{a}\right|^2+\left|\vec{b}\right|^2\), if and only if \(\vec{a},\vec{b}\) are perpendicular, given \(\vec{a}\neq0,\vec{b}\neq0\).

Choose the correct answer in Exercises 16 to 19.

16. If θ is the angle between two vectors \(\vec{a}\) and \(\vec{b}\), then \(\vec{a}.\vec{b}\geq0\) only when

(A) \(0<\theta<\frac{\pi}{2}\)

(B) \(0\le\theta\le\frac{\pi}{2}\)

(C) \(0<\theta<\pi\)

(D) \(0\le\theta\le\pi\)

17. Let \(\vec{a}\) and \(\vec{b}\) be two unit vectors and θ is the angle between them. Then \(\vec{a}+\vec{b}\) is a unit vector if

(A) \(\theta=\frac{\pi}{4}\)

(B) \(\theta=\frac{\pi}{3}\)

(C) \(\theta=\frac{\pi}{2}\)

(D) \(\theta=\frac{2\pi}{3}\)

18. The value of \(\hat{i}.\left(\hat{j}\times\hat{k}\right)+\hat{j}.\left(\hat{i}x\hat{k}\right)+\hat{k}.\left(\hat{i}x\hat{j}\right)\) is

(A) 0 (B) -1 (C) 1 (D) 3

19. If θ is the angle between any two vectors \(\vec{a} \) and \(\vec{b}\), then \(\left|\vec{a}.\vec{b}\right|=\left|\vec{a}\times\vec{b}\right|\) when θ is equal to

(A) \(0\) (B) \(\frac{\pi}{4}\) (C) \(\frac{\pi}{2}\) (D) \(\pi\)